The zero-field transverse susceptibility of a one-dimensional random bond Ising model with alternating spin 1/2 and S is exactly obtained. The integral form of the susceptibility is written. The thermal curve of the susceptibility becomes irregular when weak interactions appear with a small probability. The leading-order terms of an expansion, in terms of the moment of the distribution of a random variable, are obtained. The first-order term vanishes and the second-order term is written explicitly. The expansion up to infinite order is exactly obtained in the case of uniform spin S = 1/2. The classical limit S → ∞ is also considered. It is found that the low-temperature limit T → 0 of the susceptibility is independent of S. © 2013 IOP Publishing Ltd.